Abstract: The Lovasz Local Lemma EL75 is a powerful tool to non-constructively provethe existence of combinatorial objects meeting a prescribed collection ofcriteria. In his breakthrough paper Bec91, Beck demonstrated that aconstructive variant can be given under certain more restrictive conditions.Simplifications of his procedure and relaxations of its restrictions weresubsequently exhibited in several publications Alo91, MR98, CS00, Mos06,Sri08, Mos08. In Mos09, a constructive proof was presented that works undernegligible restrictions, formulated in terms of the Bounded OccurrenceSatisfiability problem. In the present paper, we reformulate and improve uponthese findings so as to directly apply to almost all known applications of thegeneral Local Lemma.